Hybrid magnonics in hybrid perovskite antiferromagnets

Hybrid magnonic systems are a newcomer for pursuing coherent information processing owing to their rich quantum engineering functionalities. One prototypical example is hybrid magnonics in antiferromagnets with an easy-plane anisotropy that resembles a quantum-mechanically mixed two-level spin system through the coupling of acoustic and optical magnons. Generally, the coupling between these orthogonal modes is forbidden due to their opposite parity. Here we show that the Dzyaloshinskii–Moriya-Interaction (DMI), a chiral antisymmetric interaction that occurs in magnetic systems with low symmetry, can lift this restriction. We report that layered hybrid perovskite antiferromagnets with an interlayer DMI can lead to a strong intrinsic magnon-magnon coupling strength up to 0.24 GHz, which is four times greater than the dissipation rates of the acoustic/optical modes. Our work shows that the DMI in these hybrid antiferromagnets holds promise for leveraging magnon-magnon coupling by harnessing symmetry breaking in a highly tunable, solution-processable layered magnetic platform.


I. DMI and determination of orientation of D
The Dzyaloshinskii-Moriya interaction is subject to certain symmetry constraints based on its antisymmetric nature. Moriya has quantified these constraints in 5 simple rules, of which we will only apply three 1 . The rules are summarized briefly, assuming sublattices joined by the line AB that is bisected by C: a. When a center of inversion is located at C, 0. b. When a mirror plane perpendicular to AB passes through C, ⊥ . c. When a twofold rotation axis perpendicular to AB passes through C, ⊥ . Cu-EA crystallizes in the Pbca space group 2,3 with a face-centered orthorhombic structure, where the Cu atoms are located at the inversion centers. From the symmetries of the crystal structure, it is evident that there are no mirror planes or twofold rotations, but these are replaced with glide planes and screw axes. The same symmetry constraints imposed by Moriya still hold, since the added translation does not change the direction of D. Only two symmetry elements are relevant here, the glide plane and screw axis which are oriented at the height of the unit cell along the c-axis and bisect the distance between the sublattices. Additionally, there are no inversion centers located between the two sublattices, so the DMI is allowed to exist. If it does exist, it must be ⊥ to the b-axis due to the screw axis, and ⊥ to the c-axis due to the glide plane 4,5 . Therefore, the only choice for the orientation of D is along the a-axis.

II. Spin configuration of Cu-EA under an applied field
Previously, magnetic susceptibility studies on Cu-EA have shown weak uniaxial anisotropy along a preferred in-plane direction, the a-axis. For the rest of the supplementary material, we will use (x, y, z) instead of (a, b, c), in order to be consistent with the main text. At zero applied field, the magnetization of each sublattice lies along the x axis (easy axis), and the DMI has no effect on the magnetization orientation since it is parallel to each magnetization ( 0). As the strength of the magnetic field along the x axis is increased below a certain value, the spin configuration remains unchanged. However, once the field strength reaches a certain value ~ 30 the spin flop transition will occur, and the magnetization of each sublattice will reorient along the y axis in a first order transition. At this point, the covert spin canting from the DMI becomes overt and the antisymmetric exchange becomes relevant. Here, the symmetry is broken under . As the magnetic field strength is further increased, the sublattice magnetizations begin to tilt toward the x direction with angle until they become fully aligned with the a-axis at | | 2 . When the magnetic field is applied along the y axis, no spin flop transition occurs. Instead, there is a canting of the magnetization towards the y axis with angle which occurs simultaneously with a tilting of one sublattice toward z and one toward -z. Here, the symmetry is preserved under (rotation is performed about the applied field, which is in the y-direction). At | | 2 the sublattice magnetizations are nearly aligned with the applied field except for some small tilting in the z-direction. As the field strength is increased, the magnetic moment can be further increased even past the expected saturation field. These processes are described in Figure S3.

III. Analytical LLG model with the DMI
In order to derive the analytic model for the LLG equations in the main text, the assumption that the magnetization of each sublattice can be treated in the macrospin approximation is made. The free energy per magnetic moment, then, is written as: is the external applied field, and are magnetization of each sublattice, is the exchange field, D is the DMI vector, and and are parameters relating to the easyplane and uniaxial anisotropy, respectively. Due to the fitting of the LLG models, as well as previously reported values 6 215 and 7 . Since we are interested in the physics at and above the crossing point (~100mT) we ignore the uniaxial anisotropy in our analytical equations. Moreover, the easy plane anisotropy is much stronger than the weak uniaxial anisotropy so that Cu-EA is treated as an easy-plane antiferromagnet.
To obtain equation 1 of the main text, the LLG equations are derived 7 by: , , ,

III.1. Analytical LLG model for the DMI-induced anticrossing gap
Here we assert that the DMI has an effect on the equilibrium sublattice magnetization direction, which acts similarly to an oblique magnetic field to tilt the sublattice magnetizations OOP. However, we show that it also can hybridize the optical and acoustic magnons due to its time-varying effect which is present even if the equilibrium magnetization orientation is symmetric. We will refer to these effects as the "passive" and "active" DMI, respectively. In order to analyze and separate the contributions from the active and passive DMI, we will solve the coupled LLG equations under two criteria; 1) the equilibrium magnetic moments are symmetric under twofold rotation and lie IP, but the DMI is present in the coupled LLG equations (active), and 2) the DMI-induced spin canting tilts the equilibrium magnetizations OOP but this is the only source of magnon-magnon coupling (passive). In these models, we will always set ∥ and change the direction of D between the x and y directions. Because the anisotropies along the xy plane are relatively small and are only relevant at low field ranges (<25 mT), we will ignore their effects for now in order to study the physics near the crossing point (~85 mT). At zero field, the equilibrium spins will align along the y axis and tilt toward the applied field direction with angle as the field strength is increased, as illustrated in Figure S4.
First, we will study the case in which ∥ ∥ . Assuming harmonic time dependence , , , exp substituted into equation 1 of the main text and keeping linear order terms only, we obtain: where is defined. We will assume equilibrium magnetic configurations such that which preserve rotation symmetry since the canting angle is assumed small because ≪ . To obtain an equation for we will act on the second equation with and add it to the first, and for we will subtract them. Note that . We arrive at the following Here it becomes obvious that these equations are not easily decoupled due to the term. Writing them in matrix form with the basis we obtain off diagonal matrix elements as shown below: The elements of that matrix are as follows: Above, we have completed the analytic LLG model for ∥ , and now we are ready to consider ∥ , ∥ . We will follow the same methodology as above, but now we note that allows us to diagonalize the matrix equation and separate the equations for and so no magnon-magnon coupling is predicted. We include the matrix elements for ∥ for completeness: D cos cos 2 sin sin 2 2 cos cos D cos cos 2 sin sin 2 2 2 sin cos 0 0 0 0 So, the matrix elements are zero on the off-diagonals, again predicting no magnon-magnon coupling, but are nonzero on the diagonals. Finally, to quantify the contribution of for different orientations of the applied field, we plot the eigenvalues to replicate the experimental data. By setting 75 , 215 , 28 / and 11 we can accurately reproduce the experimental data as shown in Figure S5.
Based on the literature, the DMI strength in Cu-EA has previously been determined to be ~11.9 mT 8 , which is a very good fit to our experimental results. The calculated coupling strength /2 resulting from a DMI strength of 11 mT is ~0.194 GHz, which is close to the experimental value of 0.24 GHz, indicating that level repulsion is present even if the equilibrium magnetizations are symmetric, but is due to this much more subtle interaction with the DMI which only manifests in terms of the relative oscillations between the two. When 0 ∥ , there is no anticrossing gap indicating a degenerate crossing, but when the DMI is parallel to the applied field, the crossing degeneracy is forbidden, as we have expected. Therefore, we have theoretically determined the coupling strength, and shown that it is dependent on the angle between the applied field and the DMI vector • .

III.2. Analytical LLG model for the passive DMI
Above we have shown that the observed anticrossing gap is well explained by only considering the active contribution of DMI, as opposed to interpreting the DMI as a method to tilt the spins OOP and then using the theory for oblique magnetic fields. Here, we calculate the DMI-induced crossing by only considering a symmetry broken equilibrium magnetic configuration and ignoring the time-dependent contributions in the LLG equations and show that the crossing gap is reduced. The resulting matrix equation for sublattice magnetizations tilted OOP by an angle has already been determined, we simply apply the result 9 .
For these calculations, we will use the same DMI strength as the previous section, as well as the tilting angle consistent with previous studies. At zero field, the sublattice magnetizations are aligned along the x-axis, only at sufficiently large field does the first order spin flop transition occur to orient them along the y-axis. After this transition, the weak ferromagnetism manifests and the canting angle with respect to the xy plane is sin cos where ~ 4°. So we can calculate explicitly the anticrossing gap, shown in Figure S7. The strength of the magnon-magnon coupling by only considering the presence of the DMI to tilt the spins OOP is ~0.052 GHz, which is much smaller than the measured value. Clearly, then, we have shown that while the DMI does cant the spins OOP and that this OOP tilting can induce the hybridization, we must not neglect the effects of the DMI as it relates to the elliptical oscillations of the magnetization. These oscillations represent a time-dependent magnetic interaction that amplifies the effects of magnon-magnon coupling. This opens the door to begin to search for new magnonic systems with equilibrium magnetizations that are symmetric under twofold rotation symmetry but are still hybridized.
Because is relatively small, we have shown that the active contribution of the DMI dominates the magnon-magnon coupling phenomenon, instead of the passive contribution. Our fitting results using this model provide excellent results compared to previous measurements on the strength of the DMI in this material. The key feature here is the frequency of the bare optical mode ( ) at 2 . This frequency is found by subtracting 0 0

III.3. Analytical LLG model for the gap at
and calculating the determinant. When 90°, the sublattices are exactly aligned and the frequency is identically zero. However, in the rotated frame, = cos sin . The nonzero component of the DMI vector along ′ , which is orthogonal to the net magnetization direction, causes residual spin canting at 2 and results in a nonzero frequency of the optical mode. At 90°, however, the optical mode returns to a zero frequency goldstone mode, as its frequency is proportional to cos .

IV. Possible additional sources for the observed anticrossing gap
In addition to the DMI, the magnetic anisotropy landscape along the IP direction may also give additional contributions to the observed magnon-magnon coupling, which we have assumed are small when the field is applied along a principal axis of the material. This anisotropy has been previously shown to enable magnon-magnon coupling by breaking the axial symmetry between the spin up and spin down modes of a compensated ferrimagnet. Likewise, the anisotropy may also break the twofold rotation symmetry about the applied field and enable magnon-magnon coupling between the optical and acoustic modes of an easy plane antiferromagnet. The anisotropy energy is a minimum along the x-axis, the easy axis, and continuously increases as it approaches the y-axis (second easy axis). Therefore, the anisotropy is symmetric under C2 if the magnetic field is applied along either axis shown in Figures S8a or S8c. However, if the magnetic field is applied along an intermediate direction to one of these axes, the twofold rotation symmetry is broken. This will lead to a spin structure that is canted preferentially toward the easy axis and will induce magnon-magnon coupling.

V. Observation of magnon dark modes
The optical mode in the transverse pumping configuration, as well as the optic mode above | | 2 in any configuration is considered a dark mode, meaning that it cannot be accessed by the spatially uniform microwave field which is applied during our measurement. These selection rules have long since been known and are consistent if one considers the parallel alignment of the sublattices at high field. However, we will show here that the existence of the antisymmetric DMI term will convert the driving microwave field into an effective microwave field which can be used to excite these dark modes.
In Figure S13a, the illustration of the magnon 'dark' and 'bright' modes correspond to the optical and acoustic modes under parallel alignment of the sublattices. The DMI Hamiltonian H = ⋅ can be rewritten as H = • • which exposes the effective fields that each sublattice experiences due to the deflection of the conjugate sublattice. In Figure S13b, an OOP rf microwave field generates a torque on both sublattices along the same direction, denoted as for 'primary'. Through these effective fields in the Hamiltonian, the resulting motion of the sublattices results in a secondary torque which can access the dark mode, denoted as . Therefore, the spatial non-uniformity is realized in these applied torques. Additionally, these effective fields change sign at the same frequency as the driving microwave field, and are even under twofold rotation ( ). Therefore, our interpretation of these fields in terms of an effective microwave field that can access the optic mode is validated.

VI. Numeric simulation
All features in FMR spectra are in good agreement with the simulation of the precession amplitude, as shown in Figure S14. The simulation was run with MATLAB and numerically calculates the time evolution of the magnetization of the two sublattices according to LLG equation (Eq. S1), and then the precession amplitude is obtained by fast Fourier transformation. The parameters in the LLG equation are 155 is the easy-plane anisotropy, Bex = 7.5 mT is the exchange between the two sublattices, 11 is the DM vector, 7 is the uniaxial anisotropy, and is the RF field to excite the magnetization dynamics. A Gilbert damping term is also included with alpha=0.01 for the simulation.

Equation S1
.       Figure S8. a, Schematic illustration of so-called 'dark' and 'bright' modes in the quasi-ferromagnetic phase of the layered antiferromagnet when the sublattices are aligned parallel at | | 2 . b, illustration of the torque from the effective microwave field, hrf which is a product of the motion of the sublattices.

Figure S9.
Numeric LLG simulation showing anticrossing gap and high field optic mode. This simulation was performed with the magnetic field applied at a 45 degree angle to the a-axis to support the data in Figure 3 of the main text. Figure S10. Temperature dependence of the magnetic resonance, showing a reduction in the optical mode frequency as the temperature approaches 2.5 K, consistent with previously reported results 11 . All measurements are performed in the geometry described in Figure 3 of the main text.